What is the integral of (ln(2x))/(x[1+(ln(2x))^2]) ?

The integral of (ln(2x))/(x[1+(ln(2x))^2]) is 1/2 ln|1+ln^2(2x)|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (ln(2x))/(x[1+(ln(2x))^2])

\int \frac{ln ( 2 x )}{x [ 1 + ( ln ( 2 x ) ) ^{2} ]} d x

\frac{1}{2} ln 1 + ln^{2} (2 x) + C

Solution steps

\int \frac{ln ( 2 x )}{x [ 1 + ( ln ( 2 x ) ) ^{2} ]} d x

Apply u-substitution

: \int \frac{ln ( u )}{u ( 1 + ln ^{2} ( u ) )} d u

= \int \frac{ln ( u )}{u ( 1 + ln ^{2} ( u ) )} d u

Apply u-substitution

: \int \frac{1}{2 v} d v

= \int \frac{1}{2 v} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

= \frac{1}{2} ln 1 + ln^{2} (2 x)

Add a constant to the solution

= \frac{1}{2} ln 1 + ln^{2} (2 x) + C

\int \frac{13}{( x - 5 ) ( x + 8 )} d x

\int (\frac{x ^{3} - 1}{x ^{2}}) d x

\int e^{(x)^{2}} d x

\int e^{- x} + e^{- 2 x} d x

\int (arctan (x))^{2} x d x

What is the integral of (ln(2x))/(x[1+(ln(2x))^2]) ?
The integral of (ln(2x))/(x[1+(ln(2x))^2]) is 1/2 ln|1+ln^2(2x)|+C